Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,12,Mod(1,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(48.4056203753\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9063x^{4} + 16960812x^{2} - 8250452224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{8}\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 9063x^{4} + 16960812x^{2} - 8250452224 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{5} + 68591\nu^{3} - 59012492\nu ) / 90832 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 9063\nu^{3} + 14235852\nu ) / 12976 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} - 3021 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 7790\nu^{2} + 7461475 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 3021 \)
|
| \(\nu^{3}\) | \(=\) |
\( -9\beta_{3} - 7\beta_{2} + 5326\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 7790\beta_{4} + 16072115 \)
|
| \(\nu^{5}\) | \(=\) |
\( -68591\beta_{3} - 63441\beta_{2} + 34033686\beta_1 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−81.9915 | 0 | 4674.61 | −2633.41 | 0 | −16807.0 | −215359. | 0 | 215917. | ||||||||||||||||||||||||||||||||||||
| 1.2 | −39.3338 | 0 | −500.855 | 11817.9 | 0 | −16807.0 | 100256. | 0 | −464842. | |||||||||||||||||||||||||||||||||||||
| 1.3 | −28.1647 | 0 | −1254.75 | −2648.09 | 0 | −16807.0 | 93020.9 | 0 | 74582.6 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 28.1647 | 0 | −1254.75 | 2648.09 | 0 | −16807.0 | −93020.9 | 0 | 74582.6 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 39.3338 | 0 | −500.855 | −11817.9 | 0 | −16807.0 | −100256. | 0 | −464842. | |||||||||||||||||||||||||||||||||||||
| 1.6 | 81.9915 | 0 | 4674.61 | 2633.41 | 0 | −16807.0 | 215359. | 0 | 215917. | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.12.a.i | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 63.12.a.i | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.12.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 63.12.a.i | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 9063T_{2}^{4} + 16960812T_{2}^{2} - 8250452224 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(63))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 8250452224 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 67\!\cdots\!00 \)
|
| $7$ |
\( (T + 16807)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 30\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots + 35\!\cdots\!52)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 11\!\cdots\!84 \)
|
| $19$ |
\( (T^{3} + \cdots - 74\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 31\!\cdots\!04 \)
|
| $31$ |
\( (T^{3} + \cdots - 31\!\cdots\!68)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 13\!\cdots\!84)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 11\!\cdots\!04 \)
|
| $43$ |
\( (T^{3} + \cdots + 15\!\cdots\!96)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 91\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots - 23\!\cdots\!84 \)
|
| $59$ |
\( T^{6} + \cdots - 40\!\cdots\!44 \)
|
| $61$ |
\( (T^{3} + \cdots + 51\!\cdots\!40)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 49\!\cdots\!08)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 73\!\cdots\!44 \)
|
| $73$ |
\( (T^{3} + \cdots - 14\!\cdots\!16)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 62\!\cdots\!76)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots - 24\!\cdots\!56 \)
|
| $97$ |
\( (T^{3} + \cdots - 98\!\cdots\!56)^{2} \)
|
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